The above definition assumes that the players deterministically choose one of the strategies in \(\mathcal {S}_i\)  (pure strategy). A more flexible approach consists in letting each player choose a strategy with a certain probability. In this way, we introduce a new game in which the strategies available to the players are probability distributions over \(\mathcal {S}_i\) 's. The payoff is redefined as the average payoff under the probability distributions chosen by the players. A probability distribution over \(\mathcal {S}_i\) 's is said a mixed strategy for player \(i\) . A central result of game theory [1]} states that if we allow mixed strategies, then every game with a finite number of players and with a finite number of pure strategies for each player has at least one Nash equilibrium.
